Economics 312/702 -Macroeconomics: Problem Set 4
Q1. This problem considers a simplified New Keynesian model, given by the IS equation:
xt = -φ(Rt - Etπt+1) + gt
and the New Keynesian Phillips curve:
πt = κxt + βEtπt+1 + ut
Suppose the cost shock is given by:
ut = ρuut-1 + ∈tu
while the demand shock gt is i.i.d. Suppose that monetary policy is conducted by the policy rule:
Rt = aut + (1/φ)gt
where a > 0 is a constant and φ is the same constant as in the IS equation.
(a) Solve for the equilibrium levels of inflation πt and the output gap xt as a function of the shocks ut and gt. To do so, use the policy rule in the IS equation and solve for xt in terms of the other variables. Substitute your answer into the Phillips curve, and guess that πt = kut for some constant k. Then solve for k and determine πt and xt.
(b) If policymakers want to minimize expected losses, with a loss function:
Lt = ½(ωx2t + π2t)
what is the optimal value of the policy rule coefficient a?
Q2. Suppose a policymaker operates under commitment and wants to minimize the loss function:
E0∑βt(π2t + λx2t).
Inflation is determined by a New Keynesian Phillips curve:
πt = βEtπt+1 + κxt + ut,
where the cost push shock ut is serially correlated:
ut = ρut-1 + εt,
with 0 < ρ < 1 and where εt is an i.i.d. Normal random variable with Etεt+1 = 0.
(a) Suppose that the policymaker can commit implementing a rule xt = but for some b. Find the resulting equilibrium level of inflation.
(b) What is the optimal value of b, the one the minimizes loss? What are the resulting values of inflation and the output gap?
(c) Suppose the economy is also governed by an Euler equation with a demand shock:
xt = Etxt+1 - φ(Rt - Etπt+1) + gt.
Find a policy rule for setting Rt which is implements the equilibrium for the optimal rule for b you found above. Interpret your answer.
Q3. Consider a search model of the labor market in which workers have linear utility U(c) = c, all jobs are the same and pay a wage of w units of consumption, unemployed workers get b < w in unemployment benefits, the separation rate is 50% (s = 1/2), the job finding rate is 50% (p = 1/2), and agents discount the future with factor β = 1/2.
(a) The equilibrium unemployment rate occurs when flows into and out of unemployment are equal. Find the equilibrium unemployment rate.
(b) Find the values (i.e. the total expected discounted utility) of an employed worker and an unemployed worker.
(c) Instead of the scenario outlined above, suppose that the job finding rate was p = 1, but that 1/2 of jobs paid a wage w1 < b while 1/2 of jobs paid a wage w2 > b. How would this change your results?
Q4. Consider the equilibrium search model from class, but instead of simply assuming that the wage is determined by equalizing the gains from the match, suppose that the wage is determined with a bargaining power 0 ≤ φ ≤ 1:
VE - VU = (φ/1 - φ)[VF - VV].
We previously assumed φ = 0.5.
(a) How are steady state unemployment and wages affected by an increase in φ? Interpret your answer.
(b) Suppose φ = 0. What is the wage? How is the equilibrium condition rVv(E) = 0 affected? Interpret your answer.
(c) Now suppose φ = 1. What is the wage? How is the equilibrium condition rVv(E) = 0 affected? Is there any value of E for which it is satisfied? What is the steady state of the model in this case?